Loop Products and Self-Intersections

Let M be a compact, oriented manifold and LM the space of maps of the circle into M, the free loop space of M. I will give simplified, chain-level definitions for the Chas-Sullivan "loop" product and coproduct on the homology of LM. Interactions between the product and coproduct will be discussed. I will describe a new link between geometry and the loop coproduct: If a homology class X on LM has a representative with no self-intersections of order >k, then the k-fold coproduct of X is trivial. This result is sharp for spheres and projective spaces. Joint work with Nathalie Wahl. No knowledge of loop products or string topology will be assumed.

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